Japanese Journal of Mathematics

, Volume 3, Issue 1, pp 93–119 | Cite as

Generic singularities of implicit systems of first order differential equations on the plane

  • A. A. Davydov
  • G. Ishikawa
  • S. Izumiya
  • W.-Z. Sun
Original Article


For the implicit systems of first order ordinary differential equations on the plane there is presented the complete local classification of generic singularities of family of its phase curves up to smooth orbital equivalence. Besides the well-known singularities of generic vector fields on the plane and the singularities described by a generic first order implicit differential equations, there exists only one generic singularity described by the implicit first order equation supplied by Whitney umbrella surface generically embedded to the space of directions on the plane.

Keywords and phrases

system folding phase curve Clairaut system 

Mathematics Subject Classification (2000)

58K50 (primary) 58K45, 37G05, 37Jxx (secondary) 


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Copyright information

© The Mathematical Society of Japan and Springer Japan 2008

Authors and Affiliations

  • A. A. Davydov
    • 1
  • G. Ishikawa
    • 2
  • S. Izumiya
    • 2
  • W.-Z. Sun
    • 3
  1. 1.Department of MathematicsVladimir State UniversityVladimirRussia
  2. 2.Department of MathematicsHokkaido UniversitySapporoJapan
  3. 3.School of ScienceChangchun University of Science and TechnologyChangchunChina

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